EXAMPLE 1 Use similarity statements

b. Check that the ratios of corresponding side lengths are equal.

In the diagram, ∆RST ~ ∆XYZ

a. List all pairs of congruentangles.

c. Write the ratios of the corresponding sidelengths in a statement of proportionality.

EXAMPLE 1 Use similarity statements

TRZX = 25

15 = 53

c. Because the ratios in part (b) are equal,

SOLUTION

YZRSXY = ST = TR

ZX .

a. R =~ ~~ ==X, S Y T Zand

RSXY = 20

12 = 53b.

; ST = 30

18 = 53YZ ;

GUIDED PRACTICE for Example 1

SOLUTION

J =~ ~~ ==P, K Q L RandThe congruent angles are

JKPQ = KL

QR = LJRP

The ratios of the corresponding side lengths are.

L

J K

R

P Q

1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding

side lengths in a statement of proportionality.

EXAMPLE 2 Find the scale factor

Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ZYXW to FGHJ.

EXAMPLE 2 Find the scale factor

SOLUTION

STEP 1

Identify pairs of congruent angles. From the diagram, you can see that Z F, Y G, and X H.Angles W and J are right angels, so W J. So, the corresponding angles are congruent.

EXAMPLE 2 Find the scale factor

SOLUTION

STEP 2

Show that corresponding side lengths are proportional.

XWHJ

ZYFG

YXGH

WZJF

2520=

1512=

54=

2016== 5

4

54=

= 54

3024=

EXAMPLE 2 Find the scale factor

SOLUTION

The ratios are equal, so the corresponding side lengths are proportional.

So ZYXW ~ FGHJ. The scale factor of ZYXW toFGHJ is

ANSWER

54

.

EXAMPLE 3 Use similar polygons

In the diagram, ∆DEF ~ ∆MNP. Find the value of x.

ALGEBRA

EXAMPLE 3 Use similar polygons

Write proportion.

Substitute.

Cross Products Property

Solve for x.

SOLUTION

The triangles are similar, so the corresponding side lengths are proportional.

x = 15

12x = 180

MNDE

NPEF=

=129

20x

GUIDED PRACTICE for Examples 2 and 3

In the diagram, ABCD ~ QRST.

SOLUTION

STEP 1

Identify pairs of congruent angles. From the diagram, you can see that A = Q , T = D, and B = R.Angles C and S are right angles. So, all the corresponding angles are congruent.

2. What is the scale factor of QRST to ABCD ?

GUIDED PRACTICE for Examples 2 and 3

STEP 2

Show that corresponding side lengths are proportional.

QRAB

QTAD

TSDC

RSBC

510=

612=

12= 8

16=

4 x== 1

2

12=

GUIDED PRACTICE for Examples 2 and 3

The ratios are equal, so the corresponding side lengths are proportional.

ANSWER

So QRST ~ ABCD. The scale factor of QRST to ABCD 12

.

GUIDED PRACTICE for Examples 2 and 3

3. Find the value of x.

In the diagram, ABCD ~ QRST.

Write proportion.

Substitute.

Cross Products Property

Solve for x.

SOLUTIONThe triangles are similar, so the corresponding side lengths are proportional.

RSQS

BCAC=

4 6 4+ = x

12 x+

4(12 + x) = 10 xx = 8

GUIDED PRACTICE for Examples 2 and 3

ANSWER

So the value of x is 8

EXAMPLE 4 Find perimeters of similar figures

Swimming

A town is building a new swimming pool. An Olympic pool is rectangular with length 50 meters and width 25 meters. The new pool will be similar in shape, but only 40 meters long.

Find the scale factor of the new pool to an Olympic pool.

a.

EXAMPLE 4 Find perimeters of similar figures

SOLUTION

Because the new pool will be similar to an Olympic pool, the scale factor is the ratio of the lengths,

a. 40

50 = 45

Find the perimeter of an Olympic pool and the new pool.

b.

EXAMPLE 4 Find perimeters of similar figures

x150

45 = Use Theorem 6.1 to write a proportion.

x = 120 Multiply each side by 150 and simplify.

The perimeter of the new pool is 120 meters.

ANSWER

The perimeter of an Olympic pool is 2(50) + 2(25) = 150 meters. You can use Theorem 6.1 to find the perimeter x of the new pool.

b.

GUIDED PRACTICE for Example 4

4. Find the scale factor of FGHJK to ABCDE.

In the diagram, ABCDE ~ FGHJK.

The scale factor is the ratio of the

length is

ANSWER

1510 = 3

2

GUIDED PRACTICE for Example 4

5. Find the value of x.

In the diagram, ABCDE ~ FGHJK.

x 18

1015= Use Theorem 6.1 to write a

proportion.

Cross product property.

You can use the theorem 6.1 to find the perimeter of x

x = 12

SOLUTION

15 x = 18 10

GUIDED PRACTICE for Example 4

ANSWER

The value of x is 12

GUIDED PRACTICE for Example 4

6. Find the perimeter of ABCDE.

In the diagram, ABCDE ~ FGHJK.

SOLUTION

As the two polygons are similar the corresponding side lengths are similar To find the perimeter of ABCDE first find its’ side lengths.

GUIDED PRACTICE for Example 4

FG AB = FK

AEWrite Equation

Substitute

15x = 180 Cross Products Property

x = 12 Solve for x

AE = 12

To find AE

1510 = 18

x

GUIDED PRACTICE for Example 4

Write Equation

1510 = 15

ySubstitute

15y = 150 Cross Products Property

y = 10 Solve for y

ED = 10

To find ED

FG AB = KJ

ED

GUIDED PRACTICE for Example 4

FG AB = HJ

CDWrite Equation

1510 = 12

zSubstitute

15z = 120 Cross Products Property

z = 8 Solve for z

DC = 8

To find DC

GUIDED PRACTICE for Example 4

FG AB = GH

BCWrite Equation

1510 = 9

aSubstitute

15a = 90 Cross Products Property

a = 6 Solve for x

BC = 6

To find BC

GUIDED PRACTICE for Example 4

The perimeter of ABCDE = AB + BC + CD + DE + EA= 10 + 6 + 8 + 10 + 12= 46

ANSWER The perimeter of ABCDE = 46

EXAMPLE 5 Use a scale factor

In the diagram, ∆TPR ~ ∆XPZ. Find the length of the altitude PS .

SOLUTION

First, find the scale factor of ∆TPR to ∆XPZ.

TRXZ

6 + 6= 8 + 8 = 1216 = 3

4

EXAMPLE 5 Use a scale factor

Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion.

The length of the altitude PS is 15.

Write proportion.

Substitute 20 for PY.

Multiply each side by 20 and simplify.

PSPY

3 4=

PS20

3 4=

=PS 15

ANSWER

GUIDED PRACTICE for Example 5

In the diagram, ABCDE ~ FGHJK.

In the diagram, ∆JKL ~ ∆ EFG. Find the length of the median KM.

7.

GUIDED PRACTICE for Example 5

JLEG

First find the scale factor of ∆ JKL to ∆ EFG.

SOLUTION

48 + 48= 40 + 40 = 9680 = 6

5

Because the ratio of the lengths of the median in similar triangles is equal to the scale factor, you can write the following proportion.

GUIDED PRACTICE for Example 5

SOLUTION

KM = 42

KMHF = 6

5

KM35

= 65

Write proportion.

Substitute 35 for HF.

Multiply each side by 35 and simplify.